## Backward uniqueness of semigroups arising in coupled partial differential equations systems of structural acoustics.(English)Zbl 1109.35065

The problem of backward uniqueness is considered for two structural acoustic models: one of them couples a hyperbolic equation with an elastic plate equation (parabolic type) defined on its elastic wall, and the another couples the same equation with a thermoelastic plate equation (parabolic or hyperbolic-dominated type) defined on its flexible wall. These structural acoustic models read as systems of PDE are proved to generate the first order abstract Cauchy problem with the generator $$A$$ being neither the generator of an analytic semigroup nor of a group (these cases were studied earlier). In the paper the operator $$A$$ is proved to be the generator of a strongly continuous semigroup and special estimates for the resolvent of $$A$$ are shown to hold in a Banach space. Using results for such semigroups the backward uniqueness theorems for the models considered are obtained.

### MSC:

 35L20 Initial-boundary value problems for second-order hyperbolic equations 35R35 Free boundary problems for PDEs 76Q05 Hydro- and aero-acoustics 47D06 One-parameter semigroups and linear evolution equations 35L90 Abstract hyperbolic equations